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$$W_{12} = \sum_{i}\int_{1}^{2}{\bf F}_i^{(e)}\cdot d{\bf s}_i + \sum_{i\ne j}\int_{1}^{2}{\bf F}_{ji}\cdot d{\bf s}_i$$

I want to understand this equation rigorously from the mathematical point of view. What is the Euclidean space in which the path integrals happen? The forces ##{\bf F}_{i}^{(e)}## and ##{\bf F}_{ji}## are functions of how many variables?

I thought, on my first reading, that everything happens in ##{\bf R}^3## and forces are vector fields ##{\bf F}:{\bf R}^3\to {\bf R^3}##. But on second thought it seems that ##{\bf F}_{ji}## should depend on both ##{\bf r}_j## and ##{\bf r}_i##. And also on the next page under the assumption that forces are conservative they're derived from potentials: ##{\bf F}_{ji} = -\nabla_iV_{ij}##, ##{\bf F}_i^{(e)} = -\nabla_iV_i##. Attempting to understand this mysterious to me ##\nabla_i## notation leads me to think that the ##V##s are functions of all ##N## position vectors as ##3N## independent variables, and ##\nabla_i## picks the ##3## variables to vary for gradient.

But if the ##F##s and the ##V##s are functions of all ##3N## coordinate variables, what does ##d{\bf s}_i## mean exactly in the integral? And wouldn't it make e.g. ##\int_{1}^{2}{\bf F}_i^{(e)}\cdot d{\bf s}_i## a function of all the *remaining* ##3N-3## coordinates rather than a scalar?

I'm so confused! Please help me understand what's going on mathematically here:

- What the ##F##s and the ##V##s are functions of, in the most general setting;

- Following that, how are the work integrals to be understood geometrically;

- If some force is not conservative and depends, say, directly on time or on velocity of one of the particles, how does that change the meaning of the integral? What does that make ##d{\bf s}_i##?

Thank you!